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private static int vehicleCount = 0; public Vehicle( String mk, String mdl, int maxOcc, String clr ) { make = mk; model = mdl; maxOccupants = maxOcc; color = clr; speed = 0.; vehicleCount++; } public String getColor() { return color; }

obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and then assigning true to a[i][j] if and only if there exists an edge emanating from vertex vi and terminating at vertex vj. EXAMPLE 15.16 An Adjacency Matrix for a Digraph

Note that the number of true entries in an adjacency matrix for a digraph is equal to the number of edges. Also, as with undirected graphs, different orderings of the vertex set V will result in different adjacency matrices for the same digraph.

{ { { {

int[][] a;

An incidence matrix for a digraph (V, E) is a two-dimensional integer array obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and the edges E = {e0 , e1, . . . , em 1} and then assigning 1 to a[i][j] and 1 to a[j][i] if there exists an edge emanating from vertex vi and terminating at vertex vj, and assigning 0 everywhere else. EXAMPLE 15.17 An Incidence Matrix for a Digraph

return return return return { { { {

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Figure 15.17 shows an incidence matrix for the digraph in Figure 15.13 on page 292. The first row indicates that two edges emanate from vertex a and one edge terminates there. The last 1 is in the row for vertex d and the last column. The only other nonzero entry in that column is the -1 in the row for vertex c, meaning that this edge emanates from vertex d and terminates at vertex c.

make; } model; } speed; } maxOccupants; } = = = = clr; } mk; } mdl; } spd; }

An adjacency list for a digraph (V, E) is a list that contains one element for each vertex in the graph and in which each vertex list element contains a list of the edges that emanate from that vertex. This is the same as the adjacency list for a graph, except that the links are not duplicated unless there are edges going both ways between a pair of vertices.

Figure 15.18 shows the adjacency list for the digraph in Figure 15.13 on page 292. The edge list for vertex a has two elements, one for each of the two edges that emanate from a: ab and ad.

setColor( setMake ( setModel( setSpeed(

PATHS IN A DIGRAPH A walk from vertex a to vertex b in a digraph is a sequence of edges (a0 a1, a1 a2, . . . , ak 1 ak) where a0 = a and ak = b. As with undirected paths in an undirected graphs, directed paths are usually abbreviated by their Figure 15.18 An adjacency list vertex string: p = a0 a1 a2 . . . ak 1 ak. Either way, we say that the path emanates from (or starts at) vertex a and terminates at (or ends at) vertex b. A walk is closed if it terminates at the same vertex from which it emanates. A path is a walk with all distinct vertices. A cycle is a closed walk with all interval vertices distinct. EXAMPLE 15.19 Directed Paths

In the digraph of Figure 15.13, adcabdc is a walk of length 6 which is not closed. The walk abdcacda is closed, but it is not a cycle because d (and c) are repeated internal vertices. The walk dcab is a path, which is not closed. The walk cabdc is a cycle of length 4, and the walk dcad is a cycle of length 3.

clr ) mk ) mdl ) spd )

Note that different cycles may traverse the same vertices. For example, adca and cadc are different cycles in the digraph in Figure 15.13. A digraph is strongly connected if there is a path between every pair of vertices. A digraph is weakly connected if its embedded graph is connected. A digraph that is not weakly connected is said to be disconnected. EXAMPLE 15.20 Strongly Connected and Weakly Connected Digraphs

In Figure 15.19, digraph G1 is strongly connected (and therefore also weakly connected). Digraph G2 is weakly connected, but not strongly connected because there is no path that terminates at vertex x. Digraph G3 is disconnected.

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